Partial progress towards the large sieve for square moduli

This paper condenses and expands upon our recent contributions arXiv:2603.00768 and arXiv:2603.25814, which belong together, producing a single streamlined manuscript.

We have added consequences of an analogue of Hooley's hypothesis R$^{\ast}$ for short Salié sums.

The motivation for these contributions comes from the large sieve for square moduli, which presents a specialized but compelling research object.

The first part arXiv:2601.15448 of this series used geometry of numbers and established new conditional results on the large sieve for square moduli under reasonable hypotheses on additive energies of modular square roots.

In contrast, we here explore how much can be achieved unconditionally by using purely Fourier analytic techniques.

While our progress is limited, it is worthwhile to exhaust these classical techniques first before moving to other ideas.

This may be of relevance for a potential future resolution of this problem (particularly in the function field setting).

The central problem is to estimate the number of fractions with square moduli in short intervals.